Liapunov analysis on stability of cantilevered column subjected to time variable compressive follower force at open en

Plaut, R. H.

English

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S t a b i l i t y  C r i t e r i a  for a Cant i lever  Subjected 
. to a Time-Dependent Follower Force 
* 
R.  E. P l a c t  
Center for Dynamical Systems 
Brown University 
Providence, Rhode Is land 
- __s_ M- * 
This research has  been supported by t h e  Nat ional  Aeronautics and Space 
Administration under Contract No. NGR 40-002-013, The  United S t a t e s  Army 
under Contract, No, DA-31-124-AHO-D-270, and by t h e  United S-Lates A i r  For@ 
under Contra.ct No, AFoSR-693-67 
https://ntrs.nasa.gov/search.jsp?R=19690001537 2020-03-12T06:15:09+00:00Z
Introduct ion 
I n  t h i s  Note we inves t iga t e  t h e  s t a b i l i t y  of a nonconservative 
nonautonomous system, a cant i levered column subjected a t  i t s  f r e e  end 
t o  a compressive follower force  which va r i e s  with time. 
i s  assumed t o  be present .  With t h e  use of a Liapunov type of approach, 
conditions a re  obtained which guarantee asymptotic s t a b i l i t y  o r  'almost 
sure  asymptotic s t a b i l i t y  of t h e  column. 
Viscous damping 
Consider a l i n e a r  e l a s t i c  column, b u i l t - i n  a t  one end and f r e e  
a t  t h e  other.  I n  nondimensional terms, t h e  equation of motion f o r  t h e  
l a t e r a l  displacement w(x,t) i s  assumed t o  be 
where 
The x a x i s  l i e s  alofig t h e  undisturbed s t r a i g h t  column, w i t h  x = 0 
0 < x < I, t I 0, and a comma denotes p a r t i a l  d i f f e r e n t i a t i o n .  
a t  t h e  b u i l t - i n  end and x = 1 a t  the  f r e e  end. 5 (> 0) i s  t h e  damp- 
i n g  coe f f i c i en t .  The compressive force  p ( t )  i s  applied a t  t h e  f r e e  end 
and remains tangent t o  t h e  column during motion. We assume p ( t )  i s  con- 
t inuous for  t > 0. The boundary conditions a r e  
( 0 , t )  = w (1,t) = W J X X X ( l , t )  =.o (t 2 0 )  ( 2 )  > xx w(O, t )  = W J X  
and t h e  i n i t i a l  condi t ions a r e  given by 
2 
S t a b i l i t y  Analysis I 
It i s  assumed t h a t  p ( t )  i s  s t r i c t l y  s t a t iona ry  and s a t i s f i e s  
an ergodic property which insures  t h e  equal i ty  of time averages and en- 
semble averages. Then 
1 e x i s t s  w i t h  p robab i l i t y  one [ 11 
Consider t h e  func t iona l  
where w 2 = ( W  )2J v = w and 
1 XX 1 XX 1% 
Note t h a t  V = 0 i f  w = v = 0, w i t h  V > 0 ' otherwise. The t o t a l  
time de r iva t ive  of V along so lu t ions  of (1) and (2) can be wr i t ten  i n  
t h e  form 
' w i t h  t h e  use of i n t eg ra t ion  by par t s .  
For w and v not both zero consider t h e  r a t i o  t / V J  which 
- 
%lumbers i n  brackets  designate  References a t  end of Note, 
3 
we wr i te  as 
G/V = -X-n(t) 
where 
(9) 
2 2 1  
25 + v + (2-c)Svw]dx, y =-I (v + 5w)w dx. v o  3 XX x =-J [ w  v 0 ,xx 
Then 
7j/v 5 -X -t 14 IP(t ) l  0 
Using t h e  calculus of v a r i a t i o n s  we can show t h a t  X 2 X where m 
and from t h e  inequal i ty  
( 12) 
1 2  2 
]/%(v+5w)w dx] 5 0 [ w  Jxx + (vf5w) ]dx 5 V(w,v) 
J XX 
it follows t h a t  I d  5 1. Therefore 
t /v  5 -Am + 1 p( t ) l  . 
4 
. Integrat ing t h i s  expression between 0 and t y i e l d s  
' I t  
V ( t )  N o )  e x P M  -xm -I? TJoi P( I dTl1 
where V ( t )  = V[w(x,t), v (x , t ) ]  and V ( 0 )  = V[wo(x), vo(x)]. 
Now assume t h a t  
f o r  some constant & > 0, as small as desired.  Since V ( t )  h 0 it 
then follows from (4) and (14) t h a t ,  w i t h  probabi l i ty  one, V ( t )  + 0 
1 2  1 2  1 2  as t +a, and therefore  a l s o  w dx 4 9 ,  I,w dx 3 0 ,  and Iov dx + 3  
as t +a. 
sure  'asymptotic s t a b i l i t y  i n  t h e  la rge  [2]  . 
0 ,xx 
We conclude t h a t  (17) i s  a s u f f i c i e n t  conditiolz f o r  almost 
2 
The boundary of t h e  s t a b i l i t y  region defined by (15) i s  depicted 
by t h e  so l id  l i n e  i n  Fig. 1. 
~~ 
*To be e x p l i c i t  we should def ine t h e  s t a b i l i t y  i n  terms of a norm; f o r  
2 1/2 example, we could use t h e  norm p(w,v) = []:(w:xx + v )ax] 
5 
S t a b i l i t y  Analysis I1 
I n  t h i s  sec t ion  we obtain s t a b i l i t y  c r i t e r i a  i n  terms of 
max I P W l  0 
t z o  
From (13) one can show t h a t  
if 
f o r  some & > 0. Therefore (17) i s  a su f f i c i en t  condition f o r  a s y q t o t i c  
s t a b i l i t y  of t h e  calm- ( see  Fige 1). 
For E > 4,66 a stronger  condition than (17) can be obtained 
by t h e  use of a d i f f e r e n t  technique, Consider t h e  func t iona l  V a s  i n  
(9) ,, except l e t  c = 2( 1-6) where 0 < E < d 2 .  We can wr i t e  i n  
t h e  form 
where 
We now seek conditions on p ( t )  under which W 2 O o  
T h e  calculus  of va r i a t ions  can be used t o  show t h a t  
4 1 2  Tr $1 2 
S0W,,,d" 2 3- ow dx 
where w s a t i s f i e s  (2 ) ,  We then can write 
where 0 < a < 1, The integrand i n  (21) h a s  t h e  form 
(aw2 + bw v + cv 2 ) + (Aw2 + Bw w + Cw 2 ), 
Y *  9 XX 9 , XX 
2 2 which w i l l  be non-negative i f  a 2 0, A B 0, b 6 4ac, and B S 4AC. 
If t h e  optimum values of a, A/a, and 6 a re  chosen, it can be shown 
t h a t  these  conditions a re  s a t i s f i e d  i f  
t B O  
f o r  some & > O o  It then follows t h a t  W 2 0, S -256V, and 
Therefore V ( t )  -+ 0 as t -+ 03, and we conclude t h a t  (22) i s  a s u f f i c i e n t '  
condi t ion f o r  
The 
by t h e  dashed 
than (17). 
asymptotic s t a b i l i t y  i n  t h e  l a rge ,  
boundary of t h e  s t a b i l i t y  region defined by (22) i s  depicted 
l i n e  i n  Fig, 1. For > 4.66 condi t ion (22) i s  s t ronger  
Concluding Remarks 
It i s  of i n t e r e s t  t o  compare conditions (l5), (l7), and (22) 
t o  t h e  s t a b i l i t y  condition f o r  t h e  case of a constant follower force.  
If p ( t )  = po = constant and 
i s  s t a b l e  i f  and only i f  
5 = 0, Beck [ 3 3  showed t h a t  t h e  c o l m  
(24) 
2 
Po < 2,OOthr . 
For E; > 0 . (24) i s  a s u f f i c i e n t  condition f o r  asymptotic s t a b i l i t y  
[4]; it i s  a l s o  a necessary condi t ion i f  5 i s  s m a l l  enough. 
I n  conclusion, we point  out t h a t  t h e  . s t a b i l i t y  regions derived 
here and shown i n  Fig. 1 sure ly  comprise only a p a r t  of t h e  t o t a l  stab- 
i l i t y  region. 
f irst  s t a b i l i t y  c r i t e r i a  obtained f o r  a cant i levered column subjected 
t o  a time-dependent follower force.  
However it i s  believed t h a t  t hese  r e s u l t s  represent  t h e  
Acknowledgments 
The author wishes t o  thank Professor E. F. Infan te  of Brom 
Universi ty  and Professor C.S. Hsu of t h e  Universi ty  of Cal i forn ia  f o r  
many he lp fu l  discussions an6 suggestions. 
8 
'References 
E11 
E21 
[31 
E 43 
Infante,  E. F., "On t h e  S t a b i l i t y  of Some Linear Nonautonomous Ran- 
dom Systems", Journal  of Applied Mechanics, Vola 35, No, 1, Trans. 
AS&%, Vol. 90, Ser ies  E, March 1968, pp. 7-12. 
Caughey, T. K., and Gray, A. H . , J r . ,  "On t h e  Almost Sure S t a b i l i t y  
of Linear Dynamic Systems With Stochastic Coefficients", Journal  
of Applied Mechanics, Vol. 32, No. 2, Trans. ASME, Vol. 87, Series  
E, June 1965, pp. 365-372. 
Beck, M., "Die Knicklast des e i n s e i t i g  eingespannten, t angen t i a l  
gedriickten Stabes" , Zei t sch r i f t  ffir angewandte Mathematik und 
Physik, V o l .  3, 1932, ppa 225-228 and 476-477 ( e r r a t a ) .  
.. 
19 Leipholz, H., Uber den Einf luss  der  D'hpfung b e i  nichtkonservativen 
Stab i lit 5t s pr ob lemen e la s t i s c h e r  St 5b e, 33, 
1964, pp. 308-321. 
" Ingeni eur - A r  c h i v  , V o l  
9 
Fig. 1 S t a b i l i t y  Conditions 



Stability criteria for a cantilever subjected to a time-dependent follower force

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